dorsal/arxiv
View SchemaGeneralized uncertainty relations: Theory, examples, and Lorentz invariance
| Authors | Samuel L. Braunstein, Carlton M. Caves, G. J. Milburn |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9507004 |
| URL | https://arxiv.org/abs/quant-ph/9507004 |
| DOI | 10.1006/aphy.1996.0040 |
Abstract
The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter---e.g., elapsed time---may be determined via arbitrary data analysis of arbitrary measurements on $N$ identically prepared quantum systems. The limits are expressed as generalized Mandelstam-Tamm uncertainty relations, which involve the operator that generates displacements of the parameter---e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations involving the displacement and Lorentz-rotation parameters of the Poincar\'e group.
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"abstract": "The quantum-mechanical framework in which observables are associated with\nHermitian operators is too narrow to discuss measurements of such important\nphysical quantities as elapsed time or harmonic-oscillator phase. We introduce\na broader framework that allows us to derive quantum-mechanical limits on the\nprecision to which a parameter---e.g., elapsed time---may be determined via\narbitrary data analysis of arbitrary measurements on $N$ identically prepared\nquantum systems. The limits are expressed as generalized Mandelstam-Tamm\nuncertainty relations, which involve the operator that generates displacements\nof the parameter---e.g., the Hamiltonian operator in the case of elapsed time.\nThis approach avoids entirely the problem of associating a Hermitian operator\nwith the parameter. We illustrate the general formalism, first, with\nnonrelativistic uncertainty relations for spatial displacement and momentum,\nharmonic-oscillator phase and number of quanta, and time and energy and,\nsecond, with Lorentz-invariant uncertainty relations involving the displacement\nand Lorentz-rotation parameters of the Poincar\\\u0027e group.",
"arxiv_id": "quant-ph/9507004",
"authors": [
"Samuel L. Braunstein",
"Carlton M. Caves",
"G. J. Milburn"
],
"categories": [
"quant-ph"
],
"doi": "10.1006/aphy.1996.0040",
"title": "Generalized uncertainty relations: Theory, examples, and Lorentz invariance",
"url": "https://arxiv.org/abs/quant-ph/9507004"
},
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